Given a finite simple graph, we can look at the eigenfunctions f_{k}
of the Laplacian. We are interested in the nodal surfaces
{f_{k}=0}. In the case of geometric graphs, graphs for which the
unit spheres are spheres, and if the eigenfunction does not take the value 0,
our Sard lemma assures that the hyper surface {f_{k}=0} is a
d-1 dimensional graph. We visualize these discrete Chladni figures
here for 2 spheres. The number of nodal regions is bound by k as a discrete
Courant result by Fiedler showed about 50 years ago.
We are especially interested in the ground state f_{2}
(the eigenfunction f_{1} to the eigenvalue 0
is harmonic and so constant). For 2-spheres, the ground state nodal curve
is always a circle in the 2-sphere. We conjecture that for 3-sphere graphs,
the ground state nodal surface is always a 2 sphere, so that by Jordan
Brouwer, the two nodal regions are 3-balls. Even so, the nature of
eigenfunctions of the Laplacian has a long tradition starting with
Ernst Chladni 1756-1827), we understand little about eigenfunctions of the Laplacian, even the
ground state. We are not aware that this
question has even be asked in the continuum:
Is it true that for all Riemannian 3-spheres, the ground state
nodal sphere is a 2-sphere? We believe that it is true and that
investigating the question in the discrete allows us to get a feel
for the problem.
By Courant-Fiedler the ground state f has a nodal surface C = {f=0} which
divides the 3 sphere into two 2-balls A,B.
Given a function f having a
hypersurface C=C(f) dividing the graph into two region A=A(f),B=B(f).
Let c(f) = |C(f)|/(|A(f)| |B(f)| denote the Cheeger constant of f,
where |C| is the surface area rsp. volume.
Is it true that for all geometric d-graphs G, the
number c(f)^{2}/5 is a lower bound for the ground state
energy?
All Chladni figures of the Icosahedron
Click for larger pictures
All Chladni figures of a refined Icosahedron
Click for larger pictures
Some Chladni figures of an even more refined Icosahedron